Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $t = \dfrac{5q - 40}{4q} \div \dfrac{9(q - 8)}{-8} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $t = \dfrac{5q - 40}{4q} \times \dfrac{-8}{9(q - 8)} $ When multiplying fractions, we multiply the numerators and the denominators. $t = \dfrac{ (5q - 40) \times -8 } { 4q \times 9(q - 8) } $ $ t = \dfrac {-8 \times 5(q - 8)} {4q \times 9(q - 8)} $ $ t = \dfrac{-40(q - 8)}{36q(q - 8)} $ We can cancel the $q - 8$ so long as $q - 8 \neq 0$ Therefore $q \neq 8$ $t = \dfrac{-40 \cancel{(q - 8})}{36q \cancel{(q - 8)}} = -\dfrac{40}{36q} = -\dfrac{10}{9q} $